ISSN 1061-933X, Colloid Journal, 2013, Vol. 75, No. 2, pp. 214–225. © Pleiades Publishing, Ltd., 2013. Original Russian Text © A.N. Filippov, D.Yu. Khanukaeva, S.I. Vasin, V.D. Sobolev, V.M. Starov, 2013, published in Kolloidnyi Zhurnal, 2013, Vol. 75, No. 2, pp. 237–249. 214 INTRODUCTION Development of the science of the interfacial sur- face interactions and nanotechnologies is attracting attention to studying micro- and nanosized objects and systems, including porous bodies, membranes, and thin capillaries. In such studies, researchers encounter peculiar behaviors of liquids in thin layers, capillaries, and pores. It is clear that the dynamics of flows in channels bounded by solid and/or porous walls may be adequately described, provided that the physics of nonequilibrium processes that take place at a liquid–solid interface is clear. Often, the classical boundary condition of sticking (non-slipping), which has been used for more than a century [1, 2], cannot adequately describe experimental data on liquid flows in nanosized channels. Therefore, the study of the character of liquid flows near solid surfaces has now become of not only theoretical, but also great practi- cal, significance. In particular, when extracting oil reserves that are residual or difficult to extract from porous rocks, various polymers (polyelectrolytes), ionic surfactants, or gases are added to displacing flu- ids. This causes a noticeable change in the character of oil flow in micropores due to a reduction in the viscous friction in boundary layers. As a result, the coefficient of oil displacement from a porous medium may be sig- nificantly increased. The data on the hydrodynamic forces that act on molecularly smooth hydrophilic sur- faces [3–9] are adequately described by the Reynolds lubrication theory with the use of the sticking condi- tions. However, molecularly smooth surfaces are extremely rare. Commonly, surfaces are rough, porous, or rough and porous simultaneously. In prac- tice, we encounter liquid flows in channels or pores with such surfaces. They can be exemplified by flows in capillary chromatographic columns [10–12]; pro- cesses of wetting and spreading of liquids on porous surfaces [13, 14]; and tangential flows of solutions between planar walls of apparatuses covered with membranes [15–18], solid particles [19–22], proteins [23–27], and monomolecular [28, 29] or bimolecular [30] layers of polyelectrolytes. In all these cases, a liq- uid partly flows either inside a porous layer or near a rough surface. The effective velocity of a liquid flow may be reduced in these situations. At the same time, the covering of surfaces with hydrophobizing layers causes an increase in the liquid flow velocity, which is related to a slip on an interfacial surface [31, 32]. The degree of deviation from complete sticking is com- monly characterized by a parameter that is referred to as the slip length, which represents a distance such that the extrapolation of the weighted velocity profile to which yields the zero flow velocity. Brenner was the first to consider the effect of roughness on the boundary conditions [33]. He showed that, when rough spheres are flowed around, the first-order contribution of the roughness parame- ter to the drag force is only made by two harmonics. The first harmonic is just a shift in an average sphere size, and, when the roughness level is defined by some mean line, only the second harmonics remains. It is of importance that the numerical coefficient at the latter is 0.2. Thus, at a low roughness level (lower than 0.3), the contribution of the second harmonic is very small Liquid Flow inside a Cylindrical Capillary with Walls Covered with a Porous Layer (Gel) A. N. Filippov a , D. Yu. Khanukaeva a , S. I. Vasin a , V. D. Sobolev b , and V. M. Starov c a Gubkin Russian State University of Oil and Gas, Leninskii pr. 65 bld. 1, Moscow, 119991 Russia b Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, Leninskii pr. 31, Moscow, 119071 Russia c Department of Chemical Engineering, Loughborough University Ashby Rd., Loughborough, Leicestershire, LE 11 3TU, UK Received May 5, 2012 Abstract—Viscous incompressible liquid flow in a long cylindrical capillary, the internal surface of which is covered with a permeable porous layer, is studied within the frameworks of three mathematical models. In the first model, the liquid flow in the porous layer is described by the Brinkman equation; according to the second one, the presence of the porous layer is taken into account using the Navier slip boundary conditions; and, in the third model, the Navier condition is imposed on the porous layer–liquid interface, with the flow inside the porous layer being excluded. The theoretical predictions are compared with the experimental data that one of us has obtained for liquid flow rates in porous capillaries. The validity and appropriateness of the appli- cation of the proposed models are discussed. DOI: 10.1134/S1061933X13020051